Secondary design considerations for minimum bias estimation

Several authors have suggested the method of minimum bias estimation for estimating response surfaces. The minimum bias estimation procedure achieves minimum average squared bias of the fitted model without depending on the values of the unknown parameters of the true surface. The only requirement is that the design satisfies a simple estimability condition. Subject to providing minimum average squared bias, the minimum bias estimator also provides minimum average variance of ?(x) where ?(x) is the estimate of the response at the point x.

To support the estimation of the parameters in the fitted model, very little has been suggested in the way of experimental designs except to say that a full rank matrix X of independent variables should be used. This paper presents a closer look at the estimability conditions that are required for minimum bias estimation, and from the form of the matrix X, a formula is derived which measures the amount of design flexibility available. The design flexibility is termed “the degrees of freedom” of the X matrix and it is shown how the degrees of freedom can be used to decide if other design optimality criteria might be considered along with minimum bias estimation. Several examples are provided.     

Empirical likelihood estimation for linear regression models with AR(p) error terms with numerical examples

Linear regression models are useful statistical tools to analyze data sets in different fields. There are several methods to estimate the parameters of a linear regression model. These methods usually perform under normally distributed and uncorrelated errors. If error terms are correlated the Conditional Maximum Likelihood (CML) estimation method under normality assumption is often used to estimate the parameters of interest. The CML estimation method is required a distributional assumption on error terms. However, in practice, such distributional assumptions on error terms may not be plausible. In this paper, we propose to estimate the parameters of a linear regression model with autoregressive error term using Empirical Likelihood (EL) method, which is a distribution free estimation method. A small simulation study is provided to evaluate the performance of the proposed estimation method over the CML method. The results of the simulation study show that the proposed estimators based on EL method are remarkably better than the estimators obtained from CML method in terms of mean squared errors (MSE) and bias in almost all the simulation configurations. These findings are also confirmed by the results of the numerical and real data examples.     

A new class of unbiased estimators of the variance of the systematic sample mean

We propose a class of estimators of the variance of the systematic sample mean, which is unbiased under the assumption that the population follows a superpopulation model that satisfies some mild conditions. The approach is based on the separate estimation of the portion of the variance due to the systematic component of the model and that due to the stochastic component. In particular, we deal with two estimators belonging to the proposed class that are based on moving averages and local polynomials to estimate the systematic component of the model. The latter estimators are unbiased under the assumption that the population follows a linear trend and the errors are homoscedastic and uncorrelated. Through a simulation study we show that these estimators generally outperform, in terms of bias and mean square error, the usual estimator based on the first differences also when the superpopulation model departs significantly from linearity and the errors are heteroscedastic.     

Bootstrap mean squared error of a small-area EBLUP

Concerning the estimation of linear parameters in small areas, a nested-error regression model is assumed for the values of the target variable in the units of a finite population. Then, a bootstrap procedure is proposed for estimating the mean squared error (MSE) of the EBLUP under the finite population setup. The consistency of the bootstrap procedure is studied, and a simulation experiment is carried out in order to compare the performance of two different bootstrap estimators with the approximation given by Prasad and Rao [Prasad, N.G.N. and Rao, J.N.K., 1990, The estimation of the mean squared error of small-area estimators. Journal of the American Statistical Association, 85, 163–171.]. In the numerical results, one of the bootstrap estimators shows a better bias behavior than the Prasad–Rao approximation for some of the small areas and not much worse in any case. Further, it shows less MSE in situations of moderate heteroscedasticity and under mispecification of the error distribution as normal when the true distribution is logistic or Gumbel. The proposed bootstrap method can be applied to more general types of parameters (linear of not) and predictors.     

Estimation of finite population mean in stratified sampling using scrambled responses in the presence of measurement errors

This study focuses on the estimation of population mean of a sensitive variable in stratified random sampling based on randomized response technique (RRT) when the observations are contaminated by measurement errors (ME). A generalized estimator of population mean is proposed by using additively scrambled responses for the sensitive variable. The expressions for the bias and mean square error (MSE) of the proposed estimator are derived. The performance of the proposed estimator is evaluated both theoretically and empirically. Results are also applied to a real data set.     

Clinical utility estimation for assay cutoffs in early phase oncology enrichment trials

Predictive enrichment strategies use biomarkers to selectively enroll oncology patients into clinical trials to more efficiently demonstrate therapeutic benefit. Because the enriched population differs from the patient population eligible for screening with the biomarker assay, there is potential for bias when estimating clinical utility for the screening eligible population if the selection process is ignored. We write estimators of clinical utility as integrals averaging regression model predictions over the conditional distribution of the biomarker scores defined by the assay cutoff and discuss the conditions under which consistent estimation can be achieved while accounting for some nuances that may arise as the biomarker assay progresses toward a companion diagnostic. We outline and implement a Bayesian approach in estimating these clinical utility measures and use simulations to illustrate performance and the potential biases when estimation naively ignores enrichment. Results suggest that the proposed integral representation of clinical utility in combination with Bayesian methods provide a practical strategy to facilitate cutoff decision‐making in this setting. Copyright © 2015 John Wiley & Sons, Ltd.     

Existence,uniqueness and consistency of estimation of life characteristics of three-parameter Weibull distribution based on Type-II right censored data

In this paper, we propose a new method of estimation for the parameters and quantiles of the three-parameter Weibull distribution based on Type-II right censored data. The method, based on a data transformation, overcomes the problem of unbounded likelihood. In the proposed method, under mild conditions, the estimates always exist uniquely, and the estimators are also consistent over the entire parameter space. Through Monte Carlo simulations, we further show that the proposed method of estimation performs well compared to some prominent methods in terms of bias and root mean squared error in small-sample situations. Finally, two real data sets are used to illustrate the proposed method of estimation.     

Consistent estimation of parameters and quantiles of the three-parameter gamma distribution based on Type-II right-censored data

In this paper, we propose a method of estimation of parameters and quantiles of the three-parameter gamma distribution based on Type-II right-censored data. In the proposed method, under mild conditions, the estimates always exist uniquely, and the estimators have consistency over the entire parameter space. Through Monte Carlo simulations, we further show that the proposed method performs well compared with another prominent method of estimation in terms of bias and root mean-squared error in small-sample situations. Finally, two real data sets are used for illustrating the proposed method.     

Instrumental variable estimation in ordinal probit models with mismeasured predictors

Researchers in the medical, health, and social sciences routinely encounter ordinal variables such as self‐reports of health or happiness. When modelling ordinal outcome variables, it is common to have covariates, for example, attitudes, family income, retrospective variables, measured with error. As is well known, ignoring even random error in covariates can bias coefficients and hence prejudice the estimates of effects. We propose an instrumental variable approach to the estimation of a probit model with an ordinal response and mismeasured predictor variables. We obtain likelihood‐based and method of moments estimators that are consistent and asymptotically normally distributed under general conditions. These estimators are easy to compute, perform well and are robust against the normality assumption for the measurement errors in our simulation studies. The proposed method is applied to both simulated and real data. The Canadian Journal of Statistics 47: 653–667; 2019 © 2019 Statistical Society of Canada     

JAMES-STEIN RULE ESTIMATORS IN LINEAR REGRESSION MODELS WITH MULTIVARIATE-t DISTRIBUTED ERROR

This paper considers estimation of β in the regression model y =Xβ+μ, where the error components in μ have the jointly multivariate Student-t distribution. A family of James-Stein type estimators (characterised by nonstochastic scalars) is presented. Sufficient conditions involving only X are given, under which these estimators are better (with respect to the risk under a general quadratic loss function) than the usual minimum variance unbiased estimator (MVUE) of β. Approximate expressions for the bias, the risk, the mean square error matrix and the variance-covariance matrix for the estimators in this family are obtained. A necessary and sufficient condition for the dominance of this family over MVUE is also given.     

Testing for Slope Heterogeneity Bias in Panel Data Models

ABSTRACT

Standard econometric methods can overlook individual heterogeneity in empirical work, generating inconsistent parameter estimates in panel data models. We propose the use of methods that allow researchers to easily identify, quantify, and address estimation issues arising from individual slope heterogeneity. We first characterize the bias in the standard fixed effects estimator when the true econometric model allows for heterogeneous slope coefficients. We then introduce a new test to check whether the fixed effects estimation is subject to heterogeneity bias. The procedure tests the population moment conditions required for fixed effects to consistently estimate the relevant parameters in the model. We establish the limiting distribution of the test and show that it is very simple to implement in practice. Examining firm investment models to showcase our approach, we show that heterogeneity bias-robust methods identify cash flow as a more important driver of investment than previously reported. Our study demonstrates analytically, via simulations, and empirically the importance of carefully accounting for individual specific slope heterogeneity in drawing conclusions about economic behavior.     

On estimating the mean of the selected population with unknown variance

This paper compares four estimators of the mean of the selected population from two normal populations with unknown means and common but unknown variance. The selection procedure is that the population yielding the largest sample mean is selected. The four estimators considered are invariant under both location and scale transformations. The bias and mean square errors of the four estimators are computed and compared. The conclusions are close to those reported by Dahiya ‘1974’, even for small sample sizes     

Preliminary test almost unbiased two-parameter estimators with student’s t errors and conflicting test statistics

Abstract

In this paper, we consider the preliminary test approach to the estimation of the regression parameter in a multiple regression model under multicollinearity situation. The preliminary test almost unbiased two-parameter estimators based on the Wald, the Likelihood ratio, and the Lagrangian multiplier tests are given, when it is suspected that the regression parameter may be restricted to a subspace and the regression error is distributed with multivariate Student’s t errors. The bias and quadratic risk of the proposed estimators are derived and compared. Furthermore, a Monte Carlo simulation is provided to illustrate some of the theoretical results.     

A general class of chain estimators for ratio and product of two means of a finite population

This paper proposes a class of estimators for estimating ratio and product of two means of a finite population using information on two auxiliary characters. Asymptotic expression to terms of order 0(n^-1) for bias and mean square error (MSE) of the proposed class of estimators are derived. Optimum conditions are obtained under which the proposed class of estimators has the minimum MSE. An empirical study is carried out to compare the performance of various estimators of ratio with the conventional estimators.     

MODEL‐BASED DIRECT ESTIMATION OF SMALL‐AREA DISTRIBUTIONS

Much of the small‐area estimation literature focuses on population totals and means. However, users of survey data are often interested in the finite‐population distribution of a survey variable and in the measures (e.g. medians, quartiles, percentiles) that characterize the shape of this distribution at the small‐area level. In this paper we propose a model‐based direct estimator (MBDE, Chandra and Chambers) of the small‐area distribution function. The MBDE is defined as a weighted sum of sample data from the area of interest, with weights derived from the calibrated spline‐based estimate of the finite‐population distribution function introduced by Harms and Duchesne, under an appropriately specified regression model with random area effects. We also discuss the mean squared error estimation of the MBDE. Monte Carlo simulations based on both simulated and real data sets show that the proposed MBDE and its associated mean squared error estimator perform well when compared with alternative estimators of the area‐specific finite‐population distribution function.     

Linear Regression With Nested Errors Using Probability‐Linked Data

Probabilistic matching of records is widely used to create linked data sets for use in health science, epidemiological, economic, demographic and sociological research. Clearly, this type of matching can lead to linkage errors, which in turn can lead to bias and increased variability when standard statistical estimation techniques are used with the linked data. In this paper we develop unbiased regression parameter estimates to be used when fitting a linear model with nested errors to probabilistically linked data. Since estimation of variance components is typically an important objective when fitting such a model, we also develop appropriate modifications to standard methods of variance components estimation in order to account for linkage error. In particular, we focus on three widely used methods of variance components estimation: analysis of variance, maximum likelihood and restricted maximum likelihood. Simulation results show that our estimators perform reasonably well when compared to standard estimation methods that ignore linkage errors.     

A consistent method of estimation for the three-parameter lognormal distribution based on Type-II right censored data

ABSTRACT

In this paper, we propose a parameter estimation method for the three-parameter lognormal distribution based on Type-II right censored data. In the proposed method, under mild conditions, the estimates always exist uniquely in the entire parameter space, and the estimators also have consistency over the entire parameter space. Through Monte Carlo simulations, we further show that the proposed method performs very well compared to a prominent method of estimation in terms of bias and root mean squared error (RMSE) in small-sample situations. Finally, two examples based on real data sets are presented for illustrating the proposed method.     

A risk set calibration method for failure time regression by using a covariate reliability sample

Regression parameter estimation in the Cox failure time model is considered when regression variables are subject to measurement error. Assuming that repeat regression vector measurements adhere to a classical measurement model, we can consider an ordinary regression calibration approach in which the unobserved covariates are replaced by an estimate of their conditional expectation given available covariate measurements. However, since the rate of withdrawal from the risk set across the time axis, due to failure or censoring, will typically depend on covariates, we may improve the regression parameter estimator by recalibrating within each risk set. The asymptotic and small sample properties of such a risk set regression calibration estimator are studied. A simple estimator based on a least squares calibration in each risk set appears able to eliminate much of the bias that attends the ordinary regression calibration estimator under extreme measurement error circumstances. Corresponding asymptotic distribution theory is developed, small sample properties are studied using computer simulations and an illustration is provided.     

Variance estimation based on blocked 3×2 cross-validation in high-dimensional linear regression

In high-dimensional linear regression, the dimension of variables is always greater than the sample size. In this situation, the traditional variance estimation technique based on ordinary least squares constantly exhibits a high bias even under sparsity assumption. One of the major reasons is the high spurious correlation between unobserved realized noise and several predictors. To alleviate this problem, a refitted cross-validation (RCV) method has been proposed in the literature. However, for a complicated model, the RCV exhibits a lower probability that the selected model includes the true model in case of finite samples. This phenomenon may easily result in a large bias of variance estimation. Thus, a model selection method based on the ranks of the frequency of occurrences in six votes from a blocked 3×2 cross-validation is proposed in this study. The proposed method has a considerably larger probability of including the true model in practice than the RCV method. The variance estimation obtained using the model selected by the proposed method also shows a lower bias and a smaller variance. Furthermore, theoretical analysis proves the asymptotic normality property of the proposed variance estimation.     

Ordinary Least Squares and Sample-Selection Models of Health-Care Demand Monte Carlo Comparison

This article examines alternative econometric models for health-care demand estimation. The analysis compares the Rand Health Insurance Study two-part model with sample-selection model estimators in a Monte Carlo simulation experiment designed to approximate individual-level health-care demand conditions. The underlying variable distributions are taken from cross- sectional data for a Swiss 1981 population survey. Three sets of error distribution assumptions are examined—bivariate normal, normal logistic, and Cauchy. Despite theoretical concerns with the two-part model, it outperforms the sample-selection model in terms of mean squared error of parameter estimate.